If it's a Bluetooth connection, Bluetooth has to be on and connected before you start listening to the voicemail. If your car's Bluetooth went wonky, or your phone lost the pairing information somehow, you might need to start the pairing process over again. Check your car's Bluetooth setup and see if there are any settings to enable/disable phone audio.
I have a '08 MDX. My car's hands-free unit developed a problem that completely drained the car's battery. A short time before that, on one occasion, my 5S locked up, got warm and would't respond until it completely died. Now, since the hands-free unit was disabled, the phone has functioned perfectly. May not be the same problem as yours but it might aid your diagnosis. Good luck!